Sharp constants for inequalities of Poincaré type: an application of optimal control theory
© Lou; licensee Springer. 2014
Received: 24 December 2013
Accepted: 23 April 2014
Published: 29 December 2014
Sharp constants for an inequality of Poincaré type are studied. The problem is solved by using optimal control theory.
MSC:26D10, 46E35, 49K15.
Kalyabin considered in  the following problem.
In this paper, we will solve Problem () completely with the help of optimal control theory. Since the cases were solved in , we mainly consider the cases of . We have the following.
More precisely, we have the following.
When , we have the following.
2 Transmitting Problem () to optimal control problem
Our optimal control problem corresponding to Problem () is as follows.
Therefore, we can solve Problem () by solving Problem ().
3 Pontryagin’s maximum principle
We state Pontryagin’s maximum principle for optimal control problems. Symbols in this section will have similar but probably different meanings from other sections. Thus we set this part as a separate section. We will state a result given in . For simplicity, we only state it in a simple way. In other words, Lemma 3.1 below is a special case of Theorem 3.1 and Corollary 3.1 in Chapter V of .
Now, let and . A measurable function defined on with range in U is said to be a control.
Let the function be an -valued vector function on . Assume that is Borel measurable on , continuous on and continuously differentiable on .
then is called a state/trajectory corresponding to .
is a control,
is a state corresponding to ,
Denote by the set of all admissible pairs. The set is called the set of admissible controls.
We have the following.
The vector is never zero on .
- (ii)For a.e. ,
- (iii)The pointwise maximum condition holds: for almost all and all ,
The transversality condition holds: if the mapping is continuous at and , then is orthogonal to Ω.
4 Proof of Theorem 1.1
We give the following lemma first.
We turn to the proof of Theorem 1.1.
Now, we give an optimal control version of this fact.
for some constant , . That is, is bounded in . Then, by Sobolev’s imbedding theorem, is bounded and equicontinuous in .
Therefore is a solution to Problem (). We call it an optimal pair of Problem ().
Then Ω is a manifold of dimensional 2. While the state constraints (2.2) is equivalent to .
On the other hand, for any and , if we choose , then the condition (4.1) corresponding to (2.1) holds.
- (i)we have the following non-trivial condition:(4.22)
- (ii)the maximum condition:(4.23)
- (iii)the transversality condition(4.24)
Then is an m th degree polynomial.
where and is an th degree polynomial.
for some constant , .
Then Theorem 1.1 follows from (2.7). □
The above equations imply the results in  for .
On the other hand, since is obviously continuous respect to , we can certainly get from Theorem 1.1.
5 Results for some special cases
We prove Corollaries 1.2-1.6 in this section.
Finally, (1.19) and (1.20) follow from direct calculations. We get the proof. □
Finally, (1.21) and (1.22) follow from direct calculations. We get the proof. □
Proof of Corollary 1.6 First, we get (1.24) from (1.14), (1.16), (1.19), and (1.22).
Now (1.26) follows directly from (1.25). □
This work was supported in part by 973 Program (No. 2011CB808002) and NSFC (No. 11371104).
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